Question: The graph of $y = \frac{p(x)}{q(x)}$ is shown below, where $p(x)$ and $q(x)$ are quadratic.  (Assume that the grid lines are at integers.)

[asy]
unitsize(0.6 cm);

real func (real x) {
  return (2*(x - 1)/(x + 2));
}

int i;

for (i = -8; i <= 8; ++i) {
  draw((i,-8)--(i,8),gray(0.7));
  draw((-8,i)--(8,i),gray(0.7));
}

draw((-8,0)--(8,0));
draw((0,-8)--(0,8));
draw((-2,-8)--(-2,8),dashed);
draw((-8,2)--(8,2),dashed);
draw(graph(func,-8,-2.1),red);
draw(graph(func,-1.9,8),red);
filldraw(Circle((5,func(5)),0.15),white,red);

limits((-8,-8),(8,8),Crop);
[/asy]

The horizontal asymptote is $y = 2,$ and the only vertical asymptote is $x = -2.$  Find $\frac{p(3)}{q(3)}.$
Answer: Since there is a hole at $x = 5,$ both the numerator and denominator must have a factor of $x - 5.$  Since there is a vertical asymptote at $x = -2,$ we can assume that $q(x) = (x - 5)(x + 2).$

Since the graph passes through $(1,0),$ $p(x) = k(x - 5)(x - 1)$ for some constant $k,$ so
\[\frac{p(x)}{q(x)} = \frac{k(x - 5)(x - 1)}{(x - 5)(x + 2)} = \frac{k(x - 1)}{x + 2}\]for $x \neq 5.$

Since the vertical asymptote is $y = 2,$ $k = 2,$ and
\[\frac{p(x)}{q(x)} = \frac{2(x - 1)}{x + 2}\]for $x \neq 5.$  Hence,
\[\frac{p(3)}{q(3)} = \frac{2(2)}{5} = \boxed{\frac{4}{5}}.\]